Post on 21-Feb-2021
Photonic Crystals with Tunable Refraction and Dispersion
Christopher J. Summers, Curtis W. Neff and Tsuyoshi Yamashita
School of Materials Science and Engineering Georgia Institute of Technology
Atlanta, Georgia 30332-0245
SPIE 50th Annual Meeting“Tuning the Response of Photonic Crystals”
San Diego, California31st July – 5th August, 2005
Outline
• Motivation– New ways to control beam propagation in slab waveguides
– Modify Dispersion Contours thru impact of new structures & tunable materials
• 2D Photonic Crystal Band Structures– Triangular and Square lattices– Superlattices – Non-Linear Structures
• Liquid Crystal Infiltration of 2D PC• Electro-Optical (EO) materials
• 2D Superlattice Photonic Crystal Waveguides– Static; Hybrid and E/O superlattice structures;
• Summary
Optical Routing Using Photonic Crystals
• Conventional waveguide challenges– Cross coupling between adjacent waveguides– Difficulties in alignment from outside sources– Large bend radii necessary for lossless bends
• Properties of ideal waveguides– Sharp, lossless 90 degree bends– No coupling between adjacent or crossing waveguides
• Photonic crystal based on line defects– Many systems, guiding, resonators, drop line filters, etc
• Free-Space Guiding– Virtual waveguiding
• Low divergence waveguides: “Virtual Waveguides”– Free space beam steering
• Refraction based dynamic tuning
2D PC
Biased Contact
Cross section of single pixel
ITO
Metal contact
2D PC/LC
Input Output
Virtual Waveguide Interconnect System
• Advantages of “free-space” optics– No coupling– Intersections allowed– Broadband operation
• Advantages of integrated optics– Confined beams– No hermetic packaging– One lithography step
• Disadvantages– Small feature sizes required
(beam size ~15a)
1550nm 8.55µm
1555nm 9.97µm
1545nm 7.12µm
• Tunability in these structures will allow fully reconfigurable circuits• PBG modulation introduces mirrors and wavelength tuning
Beam Steering and Dispersion
• Beam Steering– Tunable refraction
• Liquid crystal infiltration• EO material waveguide
– Intersections allowed– Limited Bandwidth operation
• Dispersion– Same phenomena as refraction– Tunable sweeping of light– Mini-spectrometer!
• Advantages of integrated optics– Confined beams– No hermetic packaging– One lithography step
• Disadvantages– Small feature sizes required
(beam size ~15a)
Superlattice Photonic Crystal StructuresBased on Triangular Lattice
• Dynamic - liquid infiltration of holes• Static - holes of different diameter• Hybrid – combination of both structures• Tunable Static – large area addressing• Tunable EO Static SL – large area addressing
Triangular Lattice Dynamic Superlattice
Static Superlattice
(a) (b)
(d)
(c)
Tunable Static E/O SL
Tunable Static SL
Dynamic Hybrid SL
Relationship BetweenReal & Reciprocal Space for SL
Reciprocal Space
Γ Χ
ΜΥ
b2
b1
BZ Folding
Real Space
aa2
a1
• Alternating rows posses different property (∆r, ∆n, or both)• New unit cell definition with two holes per lattice point• New BZ representation: symmetry reduction, hexagonal becomes rectangular: BZ folding
BZ folding in Reciprocal Space
• Hexagonal BZ representation becomes rectangular due to BZ folding
Static Superlattice Photonic CrystalSuperlattice Strength: r2/r1
• Superlattice: hole radii, r1 & r2, in adjacent rows [i, j], respectively, Lattice vector a• Increasing superlattice strength accomplished by increasing ∆r• Thus, r2 decreased relative to r1.
• Strength of superlattice defined as: extra dielectric added when r2 made smaller, r2/r1 ratio
• In Si, for r2/r1=0.857, neff = 1.654 which is ∆n = 0.654 between rows of holes
Effect of SL Strength (r2/r1) on Band Structure
• Decreasing r2 increases dielectric material in structure
• Stronger effect on air bands than dielectric bands
• Shifts bands to lower frequencies
• Decreases width of PBG
• Increases band splitting
• Similar effect in dynamic superlattice when changing ∆n
TE polarization
• Evolution of a static superlattice band structure with radius ratio(a) r2/r1 = 1, (b) r2/r1 = 0.857, (c) r2/r1 = 0.571
Magnetic Field Power Distribution
TE polarization
• Degenerate modes• Same energy
density in each mode
Band 3p Band 3p
• Non-degenerate modes
• Band 3p power > 2X Band 3s
Band 3s Band 3s
Static Superlattice PC: Dependence of PBG & BS on Hole Radii Ratio
• Dependence of PBG & Band Splitting on radius ratio:– For radii of r1 between 0.30a and 0.40a– Trade off between Gap width and band splitting – Band separation strong for r2/r1 between 1.0 and 0.55, still have a PBG.
Dispersion Contours: Dependence on r2/r1
r2/r1=1.0ωn= 0.435
Γ
Υ
Υ
Χ
Μ
Μ
r2/r1=0.571ωn= 0.366
Γ
Υ
Υ
Χ
Μ
Μ
• For ∆r = 0, (r2/r1=1), BZ folding scheme straight forward: curves converge to a single point at BZ boundaries.
• Radius modulation (r2/r1<1): curves diverge/repel at BZ boundaries• Net result: relatively flat curvature in center of BZ with high curvature near BZ
boundaries
Dispersion Surfaces for First Four Bands ofSSL Structures
Band 3pBand 3s
Dispersion Contours for SSL-Structures & Spectral Dispersion Properties
• TE polarization dispersion contours for SSL structure calculated with PWE method – SL strength of 1.0 (solid line) and 0.857 (dashed line)
• FDTD method for SL strength of 0.857 (scattered dots), • Gray lines show construction lines for a beam of wn = 0.3185 incident from air • Spectral Dispersion for r2/r1 = 0.857 for range of wn with 1% spacing between
frequencies (group of lines) 2D slab waveguide structure (scattered plot)
Dynamical Tunability: Voltage BiasBand Structure Effects
• Bias of 6 V/µm• Increase n from 2.49 to 2.598
(∆n~ 0.11)• Moves bands to lower
frequencies with bias• Equifrequency line intersects
bands at different points• Dispersion surface different for
unbiased/biased cases
Γ Μ Υ Γ Χ Μ
r2/r1=0.571
Dispersion Contours: Bias Effects
r2/r1=0.571r2/r1=1.0
• Equifrequency plane cuts across two different areas of the dispersion surface
• The different areas have similar contours, but they are shifted.• Different contours results in different optical responses
– Refraction/Beam steering– Switching/Modulation– Dispersion
Different Biasing Schemes forTunable Device Structures
EO Static SL -- uniform large area bias
Inter-digitated SL – row biasing
Hybrid (LC infiltrated) SL -- uniform large area bias
Band Structure, Dispersion & Refractionfor LC Infiltrated Static Superlattice
Band structure, Dispersion contours & Refraction response of 3p band (normalized frequency of wn=0.273)
Tunability of refraction angle ∆θr ~ 55o, EO SSL (10o)∆θr ~ 55.3o, infiltrated SSL (14o)∆θr~ 54.3o, for Inter-digitated SSLTunability limited by relative flat DC
Large Area Addressed SSLr2/r1 = 0.875wn=0.273)
Dispersion Contours and Refraction forThree SSL Devices -3s Band
• For Hybrid Static superlattice, refraction changes from negative to positive with bias∆θr = 96o – of the order of 80o for other structures
• (a & d) EO superlattice, (b & e) Hybrid Static superlattice, (c & f) inter-digitated SL Case 1: larger holes unbiased (eh1 = 2.89), smaller holes biased (eh2 = 2.25). .
Dependence of Refraction Angle on LC Refractive Index & Angle of Incidence
• Hybrid large area biased SSL structure
• Band 3s• Normalized frequency of 0.264• Incident angles: 0 – 35o.• LC index 2.2 to 2.9 • Dependence of θr slower than 3p
∆θr almost twice than in 3p band
• Band 3p• Normalized frequency of 0.273• Incident angles: -10 to 35o.• LC index 2.2 to 2.9
FDTD Visualization of Refraction Behavior in SSL
• Investigation of effect coherence on refraction
Static superlattice structure
• Static SL PC surrounded by silicon• Gaussian beam: launched at incident angles
of 0 and 12o. Width 24a.• Beam steering:
• -40.5o for θi = 0 • 47.15o for θi = 12o
• SL parameters r1=0.35a and r2=0.3a• SL strength: r2/r1 = 0.875
Virtual Waveguide Interconnect System
1550nm 8.55µm
1555nm 9.97µm
1545nm 7.12µm
• Advantages of free-space optics– No coupling– Intersections allowed– Broadband operation
• Advantages of integrated optics– Confined beams– No hermetic packaging– One lithography step
• Disadvantages– Small feature sizes required
(beam size ~15a)
Beam PropertiesDivergence Angle of 2D Gaussian Beam
A Gaussian beam spreads in the paraxial approximation in an isotropic material as:
The divergence determines the coupling efficiency into the photonic crystal
02 πωλθ
=∆
=z
x
kk
2ω0
x
z
02 πωλθ
=∆
=z
x
kk
2D Photonic CrystalsSquare Lattice
• Band Diagram– Boundaries of the
band surface– Identification of
band gaps
• Allowed Wave Vector Curve– Equifrequency curves of
the band surface– Identification of
propagation effects
Γ
Μ
Χ
)(0 ωkn Χ
• Square lattice– Hole diameter– Refractive index
Advantages of Controlling Dispersion Contours
4/w0
2π/a
Isotropic Media Photonic Crystal
Collimated BeamNormal
PropagationNegative Index
Effect
∞=
∞=
z
z
µε
negativenegative
z
z
=
=
µε
• Canceling of Z-component leads to self-collimation• Effective negative index for the energy propagation obtained• PC lattice designed to produce dispersion contours with a wide range of curvatures
– Concave –produces normal propagation - a defocusing effect– Straight – produces a collimated beam guiding– Convex – produces a negative index for sub wavelength focusing
• Collimation exploits same phenomena as sub-wavelength focusing
Curvature Reversal Near Brillouin ZoneBoundaries
• Materials– Pillars – ε = 11– Matrix – ε = 2
• Square Geometry– a = 403nm– d = 0.4a = 161nm
0.28 0.26 0.24
Band GapRegion of interest
Matrix/Cladding Pillars
• Evolution from concave to flat to convexdispersion contour with increasing frequency along Τ - M direction
• Good confinement: Robust design• Dispersion contours fitted by effective
index model
FDTD Simulation of Self-Collimated Beam
• Good confinement of Gaussian beam • Beam spread decreased by an order of
magnitude or more with beam sizes as small as 5-10 λ0
• Simulation comparing isotropic silica vsself-collimating photonic crystal – Square lattice– Silicon pillars (ε=11)– Silica matrix (ε=2)– r=0.2a; a=403nm; λ=1.55µm
• Applications include:– Virtual waveguide interconnect system– Miniaturization of conventional optical
components for small beams
Matrix/Cladding Pillars
Test Structure Design for Collimation
• Principle of operation– Gaussian like input from input
waveguide– Beam spread observable from
number of lit up output waveguides
Input WaveguidePhotonic Crystal
Small spacer
Output Waveguides
• Quality requirement– Smooth surfaces (<Ls/20)– Anisotropic sidewalls (<5°)– Uniform hole sizes in photonic
crystal (<5% locally)
Fiber
Cleaved surface Region measured
Direct Top-View Measurements
Input from the fiber Propagation thru the photonic crystal
Output from the waveguides
Photonic Crystal
• Infrared camera utilized to view scattered light from the device
Test Structures of “Virtual Waveguide”
• Input waveguide, photonic crystal, fan of waveguides for analysis• Examples of photonic crystal fabrication.
Measurements of Virtual Waveguide Effect
• Test structure with no photonic crystal
Χ
• Test photonic crystal structure exhibiting “virtual waveguide”effect: with Prof. W. Park, UC
• “Virtual waveguiding” demonstrated• Recently calculated properties of LC infiltrated structure –
negative index focusing effect predicted: ~ -20• Also tune structure for different wavelengths
Summary
• Investigated Static and Dynamic Superlattice PC configurations.– Structure and index tuning introduces new modes
• Drastic changes in band structure and dispersion surface• Tunable refraction angle changes over 80o
• Static Superlattice increases control over optical properties of PCs.– Refraction at normal incidence, negative to positive refraction observed
• Hybrid superlattice enhances tunability of optical properties of PCs.– Enhances and combines properties of static and dynamic SL PCs
• Issue is beam divergence – addressed by self-collimation• Low Divergence “virtual waveguides” demonstrated
– Focus tuning predicted in these structures• Investigating ways to combine self-collimation with tuning
Acknowledgements
Research Group
Graduate Students
• Davy Gaillot
• Xudong Wang
• Swati Jain
Postdoctoral Researcher
• Elton Graugnard, Jeff KingCollaborations with ARL:
• Drs. D. Morton, E. Forsythe &S. Blomquist
• U.S. Army Research Office - MURI Contract# DAAA19-01-1-0603
• Mike Ciftan & Rich Hammond contract monitors
Thank You!!